A brute-force computation of (REF ) calls for extensive Monte-Carlo samples in the probability space associated with both \(X\)  and \(\omega \)  [1]}, which is prohibitive under expensive queries of \(S(x,\omega )\) . Therefore, we seek to develop a sampling algorithm following the sequential BED framework, where each sample is selected making use of the existing information of previous samples. Two key components in our new approach are (1) an inexpensive surrogate model based on the variational heteroscedastic Gaussian process regression (VHGPR) to approximate the heteroscedastic ItR; and (2) an optimization based on an acquisition function to provide the next-best samples with fast convergence in computing (REF ). We next describe the two components in detail in §2.2 and §2.3, followed by the overall algorithm REF .
